Analysis of Oil Droplet Deposition Characteristics and Determination of Impact State Criterion in Aero-Engine Bearing Chamber

The research of oil/air two-phase flow and heat transfer is the fundamental work of the design of lubrication and heat transfer in aero-engine bearing chamber. The determination of impact state criterion of the moving oil droplets with the wall and the analysis of oil droplet deposition characteristics are important components. In this paper, the numerical analysis model of the impact between the moving oil droplet and the wall is established by using the finite volume method, and the simulation of oil droplet impingement on the wall is carried out. Then the effects of oil droplet diameter, impact velocity, and incident angle on the characteristic parameters of impact state are discussed. The characteristic parameters include the maximum spreading length, the maximum spreading width, and the number of splashing oil droplets. Lastly the calculation results are verified through comparing with the experimental results in the literature. The results show as follows: (1) The maximum spreading width of oil droplet firstly increases and then slows down with the incident angle and the oil droplet diameter increasing; (2) when the oil droplet diameter becomes small, the influence of the incident angle on the maximum spreading length of oil droplet is obvious and vice versa; (3) with the impact velocity and diameter of oil droplet increasing, the maximum spreading width of oil droplet increases firstly and then slows down, and the maximum spreading length increased gradually; (4) the number of splashing oil droplets increases with the incident angle and impact velocity increasing; and (5) compared with the experimental data in literature, the critical dimensionless splashing coefficient Kc proposed in this paper can better distinguish the impact state of oil droplet.


Introduction
Lubrication oil is supplied to roller or ball bearings via an under-race lubrication method and then sheds into aero-engine bearing chamber in the form of oil droplets. The high-speed moving oil droplets impact with the chamber wall at different incident angles, the diameter of oil droplets is in the range of 1-500 µm, and the impingement time is only a few microseconds. Then the oil droplet or deposits on the wall surface to form oil film, or disintegrates many smaller secondary oil droplets to suspend in the bearing chamber. Thus air, oil droplet, and oil film coexist in the bearing chamber. The whole impingement process can be simplified as the impact of oil droplet with the oblique wall. Meanwhile, the real aero-engine operating conditions are complicated, and the experimental studies are limited. It is very difficult to accurately analyze the air-oil two-phase flow, and restricts the precise design of aero-engine lubrication system. So, the numerical simulation can make up the deficiency deformed oil droplet and the wall, and obtained the change rule of the flow characteristics of the oil film on the wall. Tembely et al. [20] discussed the effect of substrates' wettability on the droplet impact droplet spreading, and the evolution of spreading diameter was determined. Adenyi et al. [21] carried out numerical and experimental study of a customized shallow sump aero-engine bearing chamber with inserts to improve oil residence volume. Bristot et al. [22] identified a Volume of Fluid (VOF) approach with turbulence damping for the transient simulation of air/gas two-phase flows in bearing chamber, which improve the accuracy of bearing chamber flows modelling.
Although there are many researches on the impact between droplet and solid wall, most of them focus on spreading characteristics after the impact between droplet and solid wall. However, the determination of impact state criteria for the adhesion and splashing state of droplet and the variation of the number of secondary droplets after the impact is rarely discussed. Meanwhile, compared with the water droplet, the high viscosity coefficient makes it show inherent characteristics when an oil droplet impacts with the wall. Therefore, it is very necessary to carry out the research on the determination of impact state criteria and the deposition characteristics of moving oil droplets in the bearing chamber.
The VOF method is used to establish the numerical model of oblique collision between moving oil droplet and solid wall. The impact state and the deposition characteristic of oil droplet is calculated under the different oil droplet diameter, incident angle, and impact velocity. Single factor analysis method is employed to analyze oil droplet impact state. The determination of impact state criterion between the dimensionless splashing coefficient and oil droplet impact state is acquired. The results can be applied to the impact analysis of oil droplet and wall in aero-engine bearing chamber. And the quantitative analysis of the number of splashing secondary oil droplets could provide the initial condition for further research on coalescence and breakup of the secondary oil droplet in the bearing chamber.

The Governing Equation of Oil Droplet Impacting with the Wall
There are two kinds of fluids involved in the process of impact between oil droplet and the wall: Lubricating oil and air. Both of them are assumed incompressible fluid and no mass and momentum exchanged with each other. The mass and momentum conservation equation of each phase are as follows: where V is the fluid velocity; p is the fluid pressure; g is the gravity acceleration; ρ and µ are the average density and average dynamic viscosity of fluid respectively; F is the momentum source term generated by oil surface. VOF method [23,24] is used to trace the free interface between oil droplet and air, and the volume fraction of oil in grid cell is calculated. The oil volume fraction equation is given by where Ω is the oil volume fraction, Ω = 0 and Ω = 1 indicate that there is no oil or full oil in the grid cell respectively; 0 < Ω < 1 indicates that oil and air coexist in the gird cell at the same time.
The average density and dynamic viscosity of the fluid in the grid cell are  (2) is given by where κ is the surface curvature, κ = ∇(n/|n|), n is the normal direction of the free interface between oil droplet and air, n = ∇Ω.

Numerical Model of Oil Droplet Impacting with Wall
The schematic diagram of the impacting between oil droplet and the solid wall is shown in Figure 1. The diameter of the incident oil droplet is D, the incident angle of the oil droplet is θ, and the impacting velocity is v. The width and length of the wall are W and L respectively.
whereκ is the surface curvature, ( / ) n κ = ∇ n , n is the normal direction of the free interface between oil droplet and air, Ω = ∇ n .

Numerical Model of Oil Droplet Impacting with Wall
The schematic diagram of the impacting between oil droplet and the solid wall is shown in Figure 1. The diameter of the incident oil droplet is D , the incident angle of the oil droplet is θ , and the impacting velocity is v . The width and length of the wall are W and L respectively. The grid model of oil droplet impacting with solid wall is shown in Figure 2, and the calculation region is W × L × H = 2 mm × 2 mm × 0.4 mm. The boundary conditions of calculation are as follows: (1) In the calculation area of oil droplet, the volume fraction of oil is set as 1, the initial oil droplet velocity is the impacting velocity; (2) in the other calculation area, the volume fraction of oil is set as 0; and (3) the wall adopts the no-slip velocity boundary condition. The impact point of oil droplet and the wall is P. Hexahedral structured grids are generated in the entire computational domain. In order to ensure the calculation accuracy, W and L are divided into 200 segments respectively, and H is divided into 40 segments. The volume cell is 1 μm × 1 μm × 1 μm, and the total grid number is 1,600,000. The mesh quality is quite fine, which completely meets the computing requirement. Considering the accuracy and sharpness, the Geo-Reconstruct is employed for interface reconstruction. And the second-order upwind scheme is employed for discretization of fluid mass and momentum conservation equation. The transient solver is used to solve the equations, in which Presto algorithm is employed for the pressure term, PISO algorithm is used for coupling the pressure and velocity term: where oil density l ρ is 926 kg/m 3 , dynamic viscosity l μ is 0.007 P·s, and surface tension coefficient l σ is 0.035 N/m. Gas density g ρ is 1.225 kg/ m 3 , dynamic viscosity g μ is 1.789×10 -5 P·s.

Results and Discussion
In this paper, the deposition characteristics of oil droplet are calculated under different operating (1) In the calculation area of oil droplet, the volume fraction of oil is set as 1, the initial oil droplet velocity is the impacting velocity; (2) in the other calculation area, the volume fraction of oil is set as 0; and (3) the wall adopts the no-slip velocity boundary condition.
The impact point of oil droplet and the wall is P. Hexahedral structured grids are generated in the entire computational domain. In order to ensure the calculation accuracy, W and L are divided into 200 segments respectively, and H is divided into 40 segments. The volume cell is 1 µm × 1 µm × 1 µm, and the total grid number is 1,600,000. The mesh quality is quite fine, which completely meets the computing requirement. Considering the accuracy and sharpness, the Geo-Reconstruct is employed for interface reconstruction. And the second-order upwind scheme is employed for discretization of fluid mass and momentum conservation equation. The transient solver is used to solve the equations, in which Presto algorithm is employed for the pressure term, PISO algorithm is used for coupling the pressure and velocity term: where oil density ρ l is 926 kg/m 3 , dynamic viscosity µ l is 0.007 P·s, and surface tension coefficient σ l is 0.035 N/m. Gas density ρ g is 1.225 kg/ m 3 , dynamic viscosity µ g is 1.789 × 10 −5 P·s.

Results and Discussion
In this paper, the deposition characteristics of oil droplet are calculated under different operating condition. The impacting velocity, incident angle and oil droplet diameter chosen are derived from the literature [3, 12,19]. The specific parameters and values are shown in Table 1. The adhesion and spreading process of oil droplet in 60 µs is shown in Figure 3 when the diameter is 150 µm, the impacting velocity is 15 m/s and the incident angle is 30 • . The adhesion and spread indicates that there are no secondary oil droplets appeared after the oil droplet impacting the wall. After impacting with the solid wall, the oil droplet spread along the wall under the combined action of gravity, inertial force, surface tension, and viscous force. It can be seen from the figure that the spreading length and spreading width of the oil film are quite different, and the oil film distribution is asymmetrical in the direction W and L. The oscillation and accumulation phenomenon occurred at the bottom of the oil film, accompanied by the appearance of "dry out spots". The reason is that the force of oil droplet is uneven in the direction of spreading length and width. Due to that the incident angle is small, the effect of gravity and viscous force on the direction of spreading length are more significant, but the spread in the direction of width is only affected by viscous force.
The adhesion and spreading process of oil droplet in 60 µs is shown in Figure 4 when the diameter is 200 µm, the impacting velocity is 10 m/s and the incident angle is 60 • . Compared with Figure 3, the spread of oil droplet on the wall is quite different. Under this condition, the spread shape of oil droplet on the wall is approximately circular, and the spread is relatively uniform in the direction of length and width. The reason is that the effect of gravity on the oil droplet become small with the incident angle increasing, and surface tension and viscous force plays a leading role.  The adhesion and spreading process of oil droplet in 60 μs is shown in Figure 4 when the diameter is 200 μm, the impacting velocity is 10 m/s and the incident angle is 60°. Compared with Figure 3, the spread of oil droplet on the wall is quite different. Under this condition, the spread shape of oil droplet on the wall is approximately circular, and the spread is relatively uniform in the direction of length and width. The reason is that the effect of gravity on the oil droplet become small with the incident angle increasing, and surface tension and viscous force plays a leading role. The adhesion and spreading process of oil droplet in 60 μs is shown in Figure 4 when the diameter is 200 μm, the impacting velocity is 10 m/s and the incident angle is 60°. Compared with Figure 3, the spread of oil droplet on the wall is quite different. Under this condition, the spread shape of oil droplet on the wall is approximately circular, and the spread is relatively uniform in the direction of length and width. The reason is that the effect of gravity on the oil droplet become small with the incident angle increasing, and surface tension and viscous force plays a leading role.  The adhesion and spreading process of oil droplet in 60 µs is shown in Figure 5 when the diameter is 250 µm, the impacting velocity is 25 m/s and the incident angle is 60 • . It can be seen that the oil droplet splashed after impacting with the wall. Due to that the diameter of the oil droplet is large and the impacting velocity is high, so the momentum of the impacting oil droplet is larger. But a large incident angle reduced the flow velocity of the deposited oil film along the wall, and the propagation velocity of the internal shock wave of the oil film is faster than that of the oil film along the wall, so "small oil column" appears at the edge of the spreading oil film. With the aid of the propagation of the shock wave, the energy accumulated at the edge, and "small oil column" extends upward, thus the "small oil column" breaks up and forms the second small splashing oil droplet. Meanwhile, it can be seen that the oil film also breaks up under the action of shock wave, which results in uneven spreading thickness of oil film on the wall, and accompanies by the appearance of dry out spots. The adhesion and spreading process of oil droplet in 60 μs is shown in Figure 5 when the diameter is 250 μm, the impacting velocity is 25 m/s and the incident angle is 60°. It can be seen that the oil droplet splashed after impacting with the wall. Due to that the diameter of the oil droplet is large and the impacting velocity is high, so the momentum of the impacting oil droplet is larger. But a large incident angle reduced the flow velocity of the deposited oil film along the wall, and the propagation velocity of the internal shock wave of the oil film is faster than that of the oil film along the wall, so "small oil column" appears at the edge of the spreading oil film. With the aid of the propagation of the shock wave, the energy accumulated at the edge, and "small oil column" extends upward, thus the "small oil column" breaks up and forms the second small splashing oil droplet. Meanwhile, it can be seen that the oil film also breaks up under the action of shock wave, which results in uneven spreading thickness of oil film on the wall, and accompanies by the appearance of dry out spots. The characteristic parameters include the maximum spreading length, the maximum spreading width, and the number of splashed secondary oil droplet, which are used to characterize the change rule of the impact state between the oil droplet and the solid wall. Figure 6 shows the relationship of the maximum spreading length and the maximum spreading width of the deposited oil film with the diameter of oil droplet under different incident angles when the impacting velocity is 20 m/s. The maximum spreading width of the oil film increases with the diameter and incident angle of oil droplet increasing, and the spreading width of the oil film increases firstly and then gradually slows down. With the diameter and incident angle of oil droplet increasing, the splash phenomenon occurs after the oil droplet impacting with the wall. The relationship between the maximum spreading length of the oil film, the diameter and the incident angle of the oil droplet is complicated. When the incident angle is small, the oil droplet is significantly affected by gravity. So, the maximum spreading length of the oil film is large. However, with the diameter of oil droplet increasing, the influence of incident angle on maximum spreading length of oil film weakens. The characteristic parameters include the maximum spreading length, the maximum spreading width, and the number of splashed secondary oil droplet, which are used to characterize the change rule of the impact state between the oil droplet and the solid wall. Figure 6 shows the relationship of the maximum spreading length and the maximum spreading width of the deposited oil film with the diameter of oil droplet under different incident angles when the impacting velocity is 20 m/s. The maximum spreading width of the oil film increases with the diameter and incident angle of oil droplet increasing, and the spreading width of the oil film increases firstly and then gradually slows down. With the diameter and incident angle of oil droplet increasing, the splash phenomenon occurs after the oil droplet impacting with the wall. The relationship between the maximum spreading length of the oil film, the diameter and the incident angle of the oil droplet is complicated. When the incident angle is small, the oil droplet is significantly affected by gravity. So, the maximum spreading length of the oil film is large. However, with the diameter of oil droplet increasing, the influence of incident angle on maximum spreading length of oil film weakens.
In Figure 7, the relationship of the maximum spreading length and width of deposited oil film with the diameter of impacting oil droplet under different impacting velocity when the incident angle is 60 • and the spreading time is 60 µs. It can be seen that with the impacting velocity increasing, the maximum spreading width of oil film increases firstly and then slows down, while the maximum spreading width of oil film increases with the impacting velocity and oil droplet diameter increasing. The larger impacting velocity and kinetic energy of oil droplet, the greater remaining energy after the oil droplet overcomes the energy dissipation in the spreading process, which is conducive to the further spreading. However, due to the splashing phenomenon on the edge of spreading, the spreading width slows down. In Figure 7, the relationship of the maximum spreading length and width of deposited oil film with the diameter of impacting oil droplet under different impacting velocity when the incident angle is 60° and the spreading time is 60 μs. It can be seen that with the impacting velocity increasing, the maximum spreading width of oil film increases firstly and then slows down, while the maximum spreading width of oil film increases with the impacting velocity and oil droplet diameter increasing. The larger impacting velocity and kinetic energy of oil droplet, the greater remaining energy after the oil droplet overcomes the energy dissipation in the spreading process, which is conducive to the further spreading. However, due to the splashing phenomenon on the edge of spreading, the spreading width slows down. The trend of oil droplet impacting velocity and the number of splashed oil droplet under different incident angles is shown in Figure 8 when the oil droplet diameter is 300 μm and the spreading time is 60 μs. It can be seen that when the impacting velocity of oil droplet is low, no splash occurs after the impact between oil droplet and the wall. With the impacting velocity of oil droplet increasing, the number of splashed oil droplet increases. Obviously, the reason is that the higher impacting momentum of oil droplet with larger impacting velocity, the more splashing oil droplet produced. Meanwhile, with the incident angle increasing, more splashing oil droplets are born at the   Figure 7, the relationship of the maximum spreading length and width of deposited oil film with the diameter of impacting oil droplet under different impacting velocity when the incident angle is 60° and the spreading time is 60 μs. It can be seen that with the impacting velocity increasing, the maximum spreading width of oil film increases firstly and then slows down, while the maximum spreading width of oil film increases with the impacting velocity and oil droplet diameter increasing. The larger impacting velocity and kinetic energy of oil droplet, the greater remaining energy after the oil droplet overcomes the energy dissipation in the spreading process, which is conducive to the further spreading. However, due to the splashing phenomenon on the edge of spreading, the spreading width slows down. The trend of oil droplet impacting velocity and the number of splashed oil droplet under different incident angles is shown in Figure 8 when the oil droplet diameter is 300 μm and the spreading time is 60 μs. It can be seen that when the impacting velocity of oil droplet is low, no splash occurs after the impact between oil droplet and the wall. With the impacting velocity of oil droplet increasing, the number of splashed oil droplet increases. Obviously, the reason is that the higher impacting momentum of oil droplet with larger impacting velocity, the more splashing oil droplet produced. Meanwhile, with the incident angle increasing, more splashing oil droplets are born at the The trend of oil droplet impacting velocity and the number of splashed oil droplet under different incident angles is shown in Figure 8 when the oil droplet diameter is 300 µm and the spreading time is 60 µs. It can be seen that when the impacting velocity of oil droplet is low, no splash occurs after the impact between oil droplet and the wall. With the impacting velocity of oil droplet increasing, the number of splashed oil droplet increases. Obviously, the reason is that the higher impacting momentum of oil droplet with larger impacting velocity, the more splashing oil droplet produced. Meanwhile, with the incident angle increasing, more splashing oil droplets are born at the same impacting velocity. The oil film is more evenly stressed in the length and width direction with the incident angle increasing. And more small oil columns emerge at the edge of the oil film, then occurs fracture, so more splashing oil droplets are produced. same impacting velocity. The oil film is more evenly stressed in the length and width direction with the incident angle increasing. And more small oil columns emerge at the edge of the oil film, then occurs fracture, so more splashing oil droplets are produced.

Determination of Impact State Criterion
The determination of impact state criterion of moving oil droplet and the wall indicates that the criterion can judge whether the critical state of oil droplet splashing occurred after the impacting with the wall. According to the previous analysis, whether the splashing phenomenon occurs is depended on the geometric and motion parameters such as the diameter, impacting velocity and incident angle of oil droplet. The relevant parameters are derived from the literature [9,14]. In order to determine whether the splashing phenomenon occurred after oil droplet impacted with the wall in the bearing chamber, the single factor analysis method is employed in a wide range of parameter. The numerical simulation of impact state between oil droplet and the wall are carried out to compute the deposition and splashing of oil droplet under the combination of several influence parameters. The value of the combination of geometric and motion parameter are recorded under corresponding conditions. In order to establish the criterion for judge the critical state of moving oil droplet impacting the wall, the dimensionless splashing coefficient K is introduced, and the expression is given by where l ρ is oil droplet density; v is oil droplet impacting velocity; D is oil droplet dimeter; l σ is surface tension of oil droplet; l μ is dynamic viscosity of oil droplet. We is the Weber number. Re is Reynolds number. The distribution relationship between the combination of geometric and operating parameters and the dimensionless splashing coefficient K which correspond to the oil droplet deposition and splashing state is shown in Figure 9. In the figure, the deposition and splashing of oil droplet are divided into two parts. The dimensionless splashing coefficient corresponding to the fitted curve was known as critical dimensionless splashing coefficient, which was represented by Kc . The fitting

Determination of Impact State Criterion
The determination of impact state criterion of moving oil droplet and the wall indicates that the criterion can judge whether the critical state of oil droplet splashing occurred after the impacting with the wall. According to the previous analysis, whether the splashing phenomenon occurs is depended on the geometric and motion parameters such as the diameter, impacting velocity and incident angle of oil droplet. The relevant parameters are derived from the literature [9,14]. In order to determine whether the splashing phenomenon occurred after oil droplet impacted with the wall in the bearing chamber, the single factor analysis method is employed in a wide range of parameter. The numerical simulation of impact state between oil droplet and the wall are carried out to compute the deposition and splashing of oil droplet under the combination of several influence parameters. The value of the combination of geometric and motion parameter are recorded under corresponding conditions. In order to establish the criterion for judge the critical state of moving oil droplet impacting the wall, the dimensionless splashing coefficient K is introduced, and the expression is given by where We = ρ l v 2 D/σ l (8) where ρ l is oil droplet density; v is oil droplet impacting velocity; D is oil droplet dimeter; σ l is surface tension of oil droplet; µ l is dynamic viscosity of oil droplet. We is the Weber number. Re is Reynolds number. The distribution relationship between the combination of geometric and operating parameters and the dimensionless splashing coefficient K which correspond to the oil droplet deposition and splashing state is shown in Figure 9. In the figure, the deposition and splashing of oil droplet are divided into two parts. The dimensionless splashing coefficient corresponding to the fitted curve was known as critical dimensionless splashing coefficient, which was represented by Kc. The fitting relationship between the dimensionless splashing coefficient and the incident angle of oil droplet obtained by univariate nonlinear regression analysis, which is given by Processes 2020, 8, x FOR PEER REVIEW 10 of 13 relationship between the dimensionless splashing coefficient and the incident angle of oil droplet obtained by univariate nonlinear regression analysis, which is given by The significance of the critical dimensionless splashing coefficient is that: (1) when K Kc < , the moving oil droplet will deposit after impacting with the solid wall, and no secondary oil droplet will be produced; (2) when K Kc > , the moving oil droplet will splash after impacting with the wall, and secondary oil droplet will be produced. The critical dimensionless splashing coefficient can be used as a criterion to judge the impact state between the oil droplet and the wall in the aero-engine bearing chamber.

Verification and Comparison of Impact State between Oil Droplet and the Wall
The dimensionless splashing coefficient can be used as the judgment criterion for the impact state of moving oil droplet and solid wall. In order to verify the rationality and validity, the critical dimensionless splashing coefficient and the judgment results provided in this paper are compared with the relevant experimental results in the literature [2,6,26], as shown in Table 2. Through comparison, it is found that the results of this paper are consistent with experimental results in the literature, and the critical dimensionless splashing coefficient proposed in this paper can better distinguish the impact state of oil droplet. The criterion for determining the impact state between moving oil droplet and the wall can be applied to the aero-engine bearing chamber, which has not been achieved in the existing research work. 623.14 splashing Figure 9. Distribution relationship between oil droplet deposition or splashing state and dimensionless splashing coefficient K.
The significance of the critical dimensionless splashing coefficient is that: (1) when K < Kc, the moving oil droplet will deposit after impacting with the solid wall, and no secondary oil droplet will be produced; (2) when K > Kc, the moving oil droplet will splash after impacting with the wall, and secondary oil droplet will be produced. The critical dimensionless splashing coefficient can be used as a criterion to judge the impact state between the oil droplet and the wall in the aero-engine bearing chamber.

Verification and Comparison of Impact State between Oil Droplet and the Wall
The dimensionless splashing coefficient can be used as the judgment criterion for the impact state of moving oil droplet and solid wall. In order to verify the rationality and validity, the critical dimensionless splashing coefficient and the judgment results provided in this paper are compared with the relevant experimental results in the literature [2,6,26], as shown in Table 2. Through comparison, it is found that the results of this paper are consistent with experimental results in the literature, and the critical dimensionless splashing coefficient proposed in this paper can better distinguish the impact state of oil droplet. The criterion for determining the impact state between moving oil droplet and the wall can be applied to the aero-engine bearing chamber, which has not been achieved in the existing research work.

Conclusions
(1) The influence of incident angle on the maximum spreading length of oil droplet is large when the diameter of oil droplet is small; the influence of incident angle on the maximum spreading length of oil droplet is small when the diameter of the oil droplet is large; while the maximum spreading width of the oil droplet increases firstly and then slows down with the incident angle increasing. (2) With the oil droplet diameter and impacting velocity increasing, the maximum spreading width increasing firstly and then slows down, while the maximum spreading length shows an increasing trend. (3) With impacting velocity and incident increases, the number of splashing oil droplet presents an increasing trend under the condition of splashing caused by oil droplet impacting with the wall. (4) Compared with the physical experiments in the literature, the rationality and validity of the critical dimensionless splashing coefficient proposed in this paper is verified. It is shown that the dimensionless splashing coefficient is feasible as a criterion for judging the impact between the moving oil droplet and the solid wall in the bearing chamber.

Conflicts of Interest:
The authors declare no conflicts of interest.
Nomenclature ρ average density of fluid µ average dynamic viscosity of fluid ρ l oil density ρ g gas density µ l oil dynamic viscosity σ l oil surface tension coefficient µ g air dynamic viscosity θ incident angle of oil droplet κ surface curvature Ω oil volume fraction p fluid pressure D diameter of oil droplet F momentum source term generated by oil surface g gravity acceleration n normal direction of the free interface between oil droplet and air v impacting velocity of oil droplet We Weber number Re Reynolds number K dimensionless splashing coefficient Kc Critical dimensionless splashing coefficient